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I'm trying to understand the definition of the integration with respect to a vector and its relation with Lebesgue integration.

In single variable calculus with $x$ is a number the integral $\int f(x) dx$ is easily to understand.

However, I have come across many situations where I met the notation $\int f(x) dx$, but now $x = (x_1, ...,x_N)$ which is a vector of dimension N. In other words, the part $d$ (i.e. "with respect to") in this case is "with respect to a vector".

I also have some knowledge of Lebesgue integral to interpret the notation $\int f d\mu$ as "integrate f with respect to the measure $\mu$". In this case, $\mu$ is the measure.

I have 2 questions please:

  1. What is the rigorous definition of the notation $\int f(x) dx$ where $x$ is a vector of N dimension and $f$ is a real valued function ? What does it mean by saying "with respect to a vector" in this case ?

  2. If I want to interpret the above notation $f(x) dx$ with respect to Lebesgue integral, what is the measure I shoud use ?

Thank you so much for your help!

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For a function $f:\Bbb{R}^n\to\Bbb{C}$, the notation $\int f(x)\,dx$ means (if nothing else is specified) $\int_{\Bbb{R}^n}f\,d\lambda_n\equiv \int_{\Bbb{R}^n}f(x)\,d\lambda_n(x)$, where $\lambda_n$ is the Lebesgue measure on $\Bbb{R}^n$; so of course for this to make sense one should assume that the function is integrable: $f\in L^1(\Bbb{R}^n, \mathcal{L}_{\Bbb{R}^n},\lambda_n)\equiv L^1(\Bbb{R}^n)$. It is also common to omit the subscript $n$ in $\lambda_n$, because context usually tells us the intended dimension.

We write $\int f(x)\,dx$ or $\int_{\Bbb{R}^n}f(x)\,dx$ simply for convenience, because on $\Bbb{R}^n$, if nothing else is suggested one always uses Lebesgue measure. Sometimes, you might see this written as $\int_{\Bbb{R}^n}f(x)\,d^nx$, where the $d^nx$ is used to stand for the integration with respect to $n$-dimensional Lebesgue measure. For concrete calculations such as in 2 or 3 dimensions, you may see the notations \begin{align} \int_{\Bbb{R}^3}f(x,y,z)\,d\lambda(x,y,z)\quad\text{or}\quad\int_{\Bbb{R}^3}f(x,y,z)\,d(x,y,z) \quad \text{or}\quad\int_{\Bbb{R}^3}f(x,y,z)\,dx\,dy\,dz \end{align} The first two aren't as common, I've seen the third a lot (sometimes the domain of integration is also left implicit). As a super concrete example, let $f:\Bbb{R}^2\to\Bbb{R}$ be $f(x,y)=e^{-(x^2+y^2)}$. Rather than writing $\int_{\Bbb{R}^2}f\,d\lambda_2$, one typically just writes $\int_{\Bbb{R}^2}e^{-(x^2+y^2)}\,dx\,dy$ or $\int_{\Bbb{R}^2}e^{-(x^2+y^2)}\,d(x,y)$.

The notation does not mean integration with respect to a vector.

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